Fraction Basics: The Meaning of the Denominator

Today’s standards place a strong emphasis on unit fractions (1/b) when students are first introduced to fractions. A unit fraction simply means the numerator is 1—it represents one part of a whole that has been divided into b equal parts. So fractions like 1/4, 1/6, and 1/20 are all unit fractions. Along with that, students are expected to develop a clear understanding of what the numerator and denominator actually represent.

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One way to build this understanding is through a classic Marilyn Burns game called Cover-Up, from her book About Teaching Mathematics. While playing the game with third graders, I used the math talk that naturally comes up during gameplay to focus on the meaning of the denominator.

We used plastic fraction tile kits to play, but it’s easy to create a set of fraction tiles by folding paper strips. You can also download a free set of printable fraction tiles from this file.

You’ll also need a fraction cube with faces labeled to match the pieces. Our cube was labeled 1/2, 1/3, 1/4, 1/6, 1/6, 1/12, and 1/12. Notice that the smaller fractions appear more often. That increases the chances students will roll pieces that fit on the whole. If you don’t have a fraction cube, you can make one using a blank cube or simply place fraction labels on the faces of a standard number cube.

To summarize the game, players try to cover the whole using smaller unit fractions. The goal is to cover it exactly—no overlaps allowed.

Start by laying out the red whole piece (the tile labeled 1). I like to have students draw a number line under the fraction bar with a dry-erase marker to connect the idea that the whole represents the distance from 0 to 1 on the number line. That concept can be surprisingly tricky for students to grasp, and I’ve written more about it here.

Next, roll the fraction cube. If the roll is 1/3, place the 1/3 tile on top of the whole. This is where the math talk comes in. Each turn, students are expected to do two things:

1. Read the fraction.
2. Explain the meaning of the denominator.

For a roll of 1/3, it might sound like this:

“I have one-third. The denominator of 3 means the whole is divided into three equal parts.”

That explanation may seem straightforward, but many students need support to say it clearly. Writing the sentence frame on the board gives them a helpful reference as they play.

Everyone followed that process for the first round of turns. When the game came back around, I introduced the next level of math talk. This time, after placing a fraction tile, students would also compare the fraction they just placed with the one before it.

For example, suppose the roll is 1/6. After placing the tile, the student still reads the fraction and explains the denominator:

“I have one-sixth. The denominator of 6 means the whole is divided into six equal parts.”

Then comes the comparison:

“1/3 is greater than 1/6 because when a whole is divided into three equal parts, the pieces are larger than when it is divided into six equal parts. More parts mean smaller pieces.”

This reasoning can be surprisingly challenging for students, but the game provides lots of natural opportunities to practice. If the next roll is 1/4, the student again reads the fraction, explains the denominator, and then compares 1/6 and 1/4.

Combining a math game with math talk put the fun in practicing a fundamental concept!